Base of a Trapezoid

Q: Does it matter which base is b₁ and which is b₂?

No, it does not matter. Because the area formula uses $b_1 + b_2$, addition is commutative, so you get the same result regardless of which base you label as $b_1$ or $b_2$.

Base of a Trapezoid

Either of the two parallel sides of a trapezoid.

Trapezoid with two parallel horizontal sides labeled "base" (top and bottom) and two diagonal sides labeled "leg

Key Formula

A = \frac{1}{2}(b_1 + b_2) \cdot h

Where:

$A$ = Area of the trapezoid
$b_1$ = Length of the first base (one parallel side)
$b_2$ = Length of the second base (the other parallel side)
$h$ = Height (altitude) — the perpendicular distance between the two bases

Worked Example

Problem: A trapezoid has bases of length 10 cm and 6 cm, and a height of 4 cm. Find its area.

Step 1: Identify the two bases and the height. The two parallel sides are the bases.

b_1 = 10 \text{ cm},\quad b_2 = 6 \text{ cm},\quad h = 4 \text{ cm}

Step 2: Add the two bases together.

b_1 + b_2 = 10 + 6 = 16 \text{ cm}

Step 3: Multiply the sum of the bases by the height.

(b_1 + b_2) \cdot h = 16 \times 4 = 64 \text{ cm}^2

Step 4: Divide by 2 to get the area.

A = \frac{1}{2} \times 64 = 32 \text{ cm}^2

Answer: The area of the trapezoid is 32 cm².

Another Example

This example works backward — given the area, height, and one base, you solve for the missing base. This is a common variation on homework and tests.

Problem: A trapezoid has an area of 54 cm² and a height of 6 cm. If one base is 12 cm long, find the length of the other base.

Step 1: Write the area formula and substitute the known values.

54 = \frac{1}{2}(12 + b_2) \cdot 6

Step 2: Simplify the right side. Half of 6 is 3.

54 = 3(12 + b_2)

Step 3: Divide both sides by 3.

18 = 12 + b_2

Step 4: Subtract 12 from both sides to isolate the unknown base.

b_2 = 18 - 12 = 6 \text{ cm}

Answer: The other base is 6 cm long.

Frequently Asked Questions

How many bases does a trapezoid have?

A trapezoid has exactly two bases. They are the pair of sides that are parallel to each other. The other two sides, which are not parallel, are called the legs of the trapezoid.

What is the difference between a base and a leg of a trapezoid?

The bases are the two parallel sides of a trapezoid, while the legs are the two non-parallel sides. The bases are used to calculate the area, and the legs connect the bases. In an isosceles trapezoid, the two legs are equal in length.

Does it matter which base is b₁ and which is b₂?

No, it does not matter. Because the area formula uses

b_1 + b_2

, addition is commutative, so you get the same result regardless of which base you label as

b_1

b_2

Base of a Trapezoid vs. Leg of a Trapezoid

	Base of a Trapezoid	Leg of a Trapezoid
Definition	Either of the two parallel sides	Either of the two non-parallel sides
How many?	Always exactly 2	Always exactly 2
Role in area formula	Both bases appear in A = ½(b₁ + b₂)·h	Legs do not appear directly in the area formula
Relationship to each other	The two bases are parallel	The two legs are generally not parallel (unless it's a parallelogram)
Special case: isosceles trapezoid	Bases are unequal in length	Both legs are equal in length

Why It Matters

You need to identify the bases of a trapezoid to calculate its area — one of the most common geometry problems in middle and high school courses. The bases also determine the midsegment (median) of a trapezoid, whose length equals the average of the two bases:

m = \frac{b_1 + b_2}{2}

. Recognizing which sides are the bases is the first step in solving nearly any trapezoid problem.

Common Mistakes

Mistake: Confusing a leg with a base and using a non-parallel side in the area formula.

Correction: Always check which two sides are parallel. Only the parallel sides are bases. The non-parallel sides are legs and do not go into the area formula.

Mistake: Using a slant side length as the height instead of the perpendicular distance between the bases.

Correction: The height (altitude) must be measured perpendicular to both bases. If you are given a leg length, you may need to use the Pythagorean theorem to find the actual height.

Related Terms

Trapezoid — The quadrilateral that has two bases
Leg of a Trapezoid — The two non-parallel sides of a trapezoid
Altitude of a Trapezoid — Perpendicular distance between the two bases
Area of a Trapezoid — Formula that directly uses both bases
Parallel Lines — Bases are parallel to each other
Side of a Polygon — Bases are specific sides of the trapezoid